In this section we will compare the cycle times of the proposed cycle and the traditional robot move cycles for 2- and 3-machine cells. Let us first consider the 2-machine case.
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Using (4.1), the cycle time of the proposed cycle with m = 2 becomes:
Tproposed(2) = 4² + 6δ + 1/2max{0, P − (2² + 4δ)}. (4.6) and using (4.2), the lower bound for the traditional robot move cycles becomes:
Tf s(2)= max{6² + 6δ + min{P, δ}, 4² + 4δ + P/2}. (4.7)
The next theorem will establish an important contribution.
Theorem 4.3 The proposed robot move cycle A01A02A13A23 gives the mini-mum cycle time for 2-machine identical parts robotic cell scheduling problem with process and operational flexibility.
Proof. A simple comparison of equations (4.6) and (4.7) for P ∈ [0, δ], P ∈ (δ, 2² + 4δ], P ∈ (2² + 4δ, 4² + 6δ] and P ∈ (4² + 6δ, ∞) yields
Tproposed(2) ≤ Tf s(2). 2
Note that the proposed cycle is not necessarily the best pure cycle. However, Theorem 4.3 proves that even this cycle dominates all of the classical robot move cycles. In a 2-machine cell there are 6 pure cycles, C1 through C6, for which the activity sequences and the cycle time values are presented in Appendix A. The following theorem compares the pure cycles with each other and determines the regions of optimality.
Theorem 4.4 If P < 2² + 4δ then C1 is optimal, if P > 2² + 4δ then C6 is optimal, if P = 2² + 4δ then both C1 and C6 perform equally well.
Proof. Observing the cycle times of the cycles presented in Appendix A, one can easily conclude that C1 dominates C2, C3, C4 and C5. A simple
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comparison of the cycle times of C1 and C6 concludes the proof. 2
Now let us consider 3-machine cells. Using (4.1), the cycle time of the proposed cycle with m = 3 becomes:
Tproposed(3)= 4² + 8δ + 1/3max{0, P − (4² + 10δ)}. (4.8)
Recall that in a 3-machine cell there are six feasible 1-unit cycles which can be listed as follows:
S13 = (A0A1A2A3), S23 = (A0A2A1A3), S33 = (A0A1A3A2), S43 = (A0A3A1A2), S53 = (A0A2A3A1), S63 = (A0A3A2A1).
The lower bound of the classical robot move cycles found in Theorem 4.1 becomes the following for 3-machine robotic cells:
Tf lowshop= max{8(² + δ) + min{P, δ}, 4² + 4δ + (P/3)}. (4.9)
The forthcoming corollary to Theorem 4.2 provides the regions where the proposed cycle is the best for 3-machine cells.
Corollary 4.1 If δ ≤ 2² or P ≤ 16² + 13δ, then the proposed cycle gives the minimum cycle time for 3-machine cells.
Now let us consider the region where the lower bound of the flowshop type robot move cycles is less than the cycle time of the proposed robot move cycle.
That is, δ > 2² and P > 16² + 13δ. First we concentrate on the 1-unit robot move cycles since they are simple, practical, easy to understand and provably optimal for 3-machine flowshop type systems. The following lemma is very useful in reducing the number of potentially optimal robot move cycles:
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Lemma 4.2 The proposed cycle dominates all flowshop type 1-unit cycles except S63.
Proof. Let us consider each 1-unit cycle one by one:
S31: For the cycle S13, whatever the allocation of the operations is, the cycle time is the same. The cycle time derived by Sethi et al. [86] is:
TS3
1(Πk)= 8² + 8δ + P.
As it is seen, the cycle time does not depend on the allocation. When we compare this cycle time with the cycle time of the proposed cycle given in (4.8), Tproposed(3) < TS13(Πk). Thus we conclude that the proposed cycle dominates S13. S32: Let us derive the cycle time of the cycle S23 considering the assumptions of this study. Consider an arbitrary allocation matrix Πk and the ith repetition of this cycle. Initially the second machine is loaded with a part having allocation type (i− 1) and the robot is in front of the input buffer. The robot takes a part from the input buffer and loads it to the first machine, (2² + δ), moves to second machine, waits if necessary for the machine to finish the processing of the part with allocation type (i− 1), (δ + w(i−1)2), unloads the second machine and loads the third machine, (2² + δ), moves to the first machine, waits if necessary for the machine to finish the processing of the part with allocation type i, (2δ + wi1), unloads the first machine and loads the second machine, (2² + δ), moves to the third machine and waits if necessary for the part with allocation type (i− 1), (δ + w(i−1)3), unloads the machine and drops the part to the output buffer, (2² + δ), returns back to input buffer, (4δ). Hence the time for the ith repetition of the cycle S23 with allocation matrix Πk becomes:
8² + 12δ + wi1+ w(i−1)2+ w(i−1)3. Let us denote max{0, a} as (a)+. With this notation,
wi1 = (Pi1− 4² − 8δ − w(i−1)2)+, w(i−1)2 = (P(i−1)2 − 2² − 4δ − w(i−2)3)+ and
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w(i−1)3 = (P(i−1)3−2²−4δ−wi1)+are the waiting times in front of the machines 1, 2 and 3, respectively. For all k repetitions, we have the following:
TS23(Πk) = 8² + 12δ + 1/k(
where W is defined to be the total waiting time in front of the three machines for the k parts produced according to the allocation matrix Πk. This yields;
2W ≥ k(P − 8² − 16δ) ⇒ W ≥ k/2(P − 8² − 16δ). are the waiting times in front of machines 2 and 3, respectively. Then we have the following:
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Pi1+ wi2+ w(i−1)3 ≥ Pi1+ Pi2− 2² − 4δ − w(i−1)3+ P(i−1)3− 4² − 6δ − Pi1. This yields:
2(Pi1+ wi2+ w(i−1)3)≥ 2(Pi1+ w(i−1)3) + wi2 ≥ Pi1+ Pi2+ P(i−1)3− 6² − 10δ.
For all k repetitions we have the following:
2
Let W be as defined previously. Then we can write the following:
2W ≥ P − 6² − 10δ ⇒ W ≥ 1/2(P − 6² − 10δ).
Then for S33 we have the following:
TS3
3(Πk) ≥ 8² + 10δ + 1/2(P − 6² − 10δ) = 1/2(P + 10² + 10δ). (4.11) S34: Total time for the ith repetition of S43 is the following:
8² + 12δ + wi1+ Pi2+ w(i−1)3,
where wi1 = (Pi1− 2² − 6δ − w(i−1)3)+ and w(i−1)3= (P(i−1)3− 2² − 6δ)+ are the waiting times in front of machines 1 and 3, respectively. A similar procedure that we used for S33 yields, W ≥ 1/2(P − 4² − 12δ) and
TS34(Πk)≥ 1/2(12² + 12δ + P ). (4.12) S35: Total time for the ith repetition of S53 is the following:
8² + 10δ + wi1+ w(i−1)2+ P(i−1)3,
where wi1= (Pi1−4²−6δ−w(i−1)2−P(i−1)3)+, and w(i−1)2 = (P(i−1)2−2²−4δ)+ are the waiting times in front of the machines 1 and 2, respectively. From here we get, W ≥ 1/2(P − 6² − 10δ) and the lower bound for S53 becomes:
TS35(Πk)≥ 1/2(10² + 10δ + P ). (4.13)
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Comparing the lower bounds for the cycles S23, S33, S43 and S53, given in equations (4.10), (4.11), (4.12) and (4.13) respectively we get the following:
TS3 with the cycle time of the proposed cycle given in (4.8):
1/2(8² + 8δ + P ) = 1/2(1/3(24² + 24δ + 3P )).
S36: Example 4.1 shows that S63 cannot be dominated by the proposed robot
move cycle. 2
1-unit cycles are important because they are simple, practical and easy to understand. Also if the system is assumed to be a flowshop then 1-unit cycles are provably optimal for 2-machine cells ([86]) and 3-machine cells ([20]).
However, Akturk et al. [2] proved that with the assumption of operational flexibility, even in 2-machines case, a 2-unit cycle can result in smaller cycle times than the 1-unit robot move cycles for some parameter ranges. This motivates us to consider the 2-unit cycles. Hall et al. [43] derived the activity sequences of all feasible 2-unit cycles in a 3-machine robotic cell. In Appendix B we present a completely new procedure to derive the activity sequences of these cycles and list them. This new procedure utilizes the fact that all 2-unit cycles are made up from two 1-unit cycles. That is, let Si3 and Sj3 be two different 1-unit cycles. Then, in a 2-unit cycle, Sij3 is simply a combination of
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Si3and Sj3; during some part of the cycle the robot follows the activity sequence of Si3 and during the remaining part of the cycle the robot follows the activity sequence of Sj3.
The following lemma derives a general lower bound, T2, for all of the 2-unit robot move cycles with any allocation matrix Πk.
Lemma 4.3 T2 = 1/2(P + 8² + 8δ) where T2 ≤ minS3ij,k{TS3ij(Π∗
k)}.
Proof. For the clarity of the presentation, we refer the reader to Appendix C
for the proof. 2
The following lemma proves that the proposed cycle dominates all flowshop type 2-unit robot move cycles.
Lemma 4.4 The proposed cycle dominates all flowshop type 2-unit cycles.
Proof. With Corollary 4.1 we assert that the proposed cycle gives the minimum cycle time for P ≤ 16² + 13δ. Now let us consider the region where P > 16²+13δ. In this region the cycle time of the proposed cycle given in (4.8) becomes 4²+8δ +1/3(P−4²−10δ) which can be rewritten as 1/3(P +8²+14δ).
When we compare this cycle time with the lower bound we found in Lemma 4.3, we have the following:
T2 = 1/2(P + 8² + 8δ) = 1/6(3P + 24² + 24δ)≥ 1/6(2P + 40² + 37δ)
= 1/3(P + 20² + (18.5)δ) > 1/3(P + 8² + 14δ) = Tproposed.
This completes the proof. 2
Until now we considered all the 1 and 2-unit cycles and showed that the proposed cycle dominates all except the 1-unit cycle S63. Knowing that the
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proposed cycle dominates all 2-unit cycles, one can conjecture that it also dominates 3 and higher unit cycles. Proving or disproving this conjecture is not so simple because the number of feasible robot move cycles increases drastically as n, the number of units produced in one cycle increases and deriving and comparing these cycles with the proposed cycle become quite complex. Additionally, the proposed cycle is simple, practical and easy to implement. Furthermore, there is no allocation problem to be solved for this cycle. More importantly, the worst case bound of the proposed cycle found in Lemma 4.1 becomes 1.08 for 3-machine robotic cells. As a result of these observations, we conclude that what little improvement we might attain (if any) by considering 3 and higher unit cycles will not be sufficient enough to justify the effort that will be spent for this purpose.
4.3 Concluding Remarks
In this chapter, we considered a new class of robot move cycles, called the pure cycles, resulting from the flexibility of the CNC machines. Since there is a huge number of such cycles in an m-machine robotic cell, we proposed one of the cycles among this class which is extensively used in industry due to its simplicity in understanding and implementation. We proved in Theorem 4.3 that this cycle, in fact, dominates the traditional robot move cycles for m = 2.
With Theorem 4.2 we found the regions where the proposed cycle dominates the traditional robot move cycles for m≥ 3. In order to prove this theorem, we compared the proposed cycle with the lower bound of the classical robot move cycles. For the remaining regions we proved that the proposed cycle dominates all of the 1-unit robot move cycles except S63 and all of the 2-unit robot move cycles in 3-machine cells. We also found a worst case performance bound of the proposed cycle with respect to the traditional robot move cycles for
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the remaining regions. Furthermore, with the reduced cycle times (increased throughput), our results enable the justification of additional tool inventories that will be incurred when loading a copy of every required tool to both of the machines (this might also necessitate a larger tool magazine). As a final remark, in the new move cycle each part is loaded and unloaded only once, which means less gaging; probably one of the important reasons why this cycle is preferred in practice. An extended version of this chapter is accepted for publication [39].